Author

zoo un park

packages used

Code 
library(tidyverse)
library(lubridate)
library(zoo)       
library(forecast)   
library(tseries)   
library(patchwork) 
library(here)
library(plotly)

Time series plot

Code 
uk = read.csv("../data/fx rate/UK_US.csv") 
uk <- uk %>% mutate(observation_date = as.Date(observation_date))%>%
  rename(UK_USD = DEXUSUK)%>%
  mutate(UK_USD = na.locf(UK_USD, na.rm = FALSE))

uk_plot <- ggplot(uk, aes(x = observation_date, y = UK_USD)) +
  geom_line(color = "blue") +
  labs(title = "GBP/USD Exchange Rate",
       x = "Date", y = "GBP(GBP per 1 USD)") +
  theme_minimal()

uk_plot

Code 
jp <- read_csv("../data/fx rate/JP_US.csv") %>%
  mutate(observation_date = as.Date(observation_date)) %>%
  rename(JP_USD = DEXJPUS,
         Date   = observation_date) %>%
  mutate( JP_USD = na.locf(JP_USD, na.rm = FALSE),
          USD_JP = 1 / JP_USD)  

jp_plot <- ggplot(jp, aes(x = Date, y = USD_JP)) +
  geom_line(color = "blue") +
  labs(title = "JPY/USD Exchange Rate ",
       x = "Date", y = "JPY(JPY per 1 USD)") +
  theme_minimal()

jp_plot

Code 
crude_oil <- read_csv(here("data/oil","oil_cleaned.csv")) %>%
  mutate(Date = as.Date(Date)) %>%
  arrange(Date) %>%
  mutate(across(-Date, ~ na.locf(.x, na.rm = FALSE)))

ggplot(crude_oil, aes(x = Date, y = WTI)) +
  geom_line(color = "darkgreen") +
  labs(title = "Crude Oil Prices", x = "Date", y = "Price") +
  theme_minimal()

Code 
sp500 <- read_csv(here("data/sp500","sp500_cleaned.csv")) %>%
  mutate(Date = as.Date(Date)) %>%
  arrange(Date) %>%
  mutate(across(-Date, ~ na.locf(.x, na.rm = FALSE)))

ggplot(sp500, aes(x = Date, y = Close)) +
  geom_line(color = "blue") +
  labs(title = "S&P 500 Index", x = "Date", y = "Level") +
  theme_minimal()

Code 
btc <- read_csv(here("data/btc","btc_cleaned.csv")) %>%
  mutate(Date = as.Date(Date)) %>%
  arrange(Date) %>%
  mutate(across(-Date, ~ na.locf(.x, na.rm = FALSE)))

ggplot(btc, aes(x = Date, y = Close)) +
  geom_line(color = "gold") +
  labs(title = "Bitcoin Price", x = "Date", y = "Price") +
  theme_minimal()

Code 
spread<- read_csv("../data/yield/merge_yield.csv")%>%
mutate(Date = as.Date(Date),
         Year = lubridate::year(Date)) %>%    
  arrange(Date)%>%
  mutate(across(ends_with("_spread"), ~ na.locf(.x, na.rm = FALSE))) %>%
  mutate(across(ends_with("_spread"), ~ na.locf(.x, fromLast = TRUE)))

jp_plot <- plot_ly(spread, x = ~Date, y = ~JP_Spread, type = 'scatter', mode = 'lines',
                   name = "Japan (10Y-2Y)") %>%
  layout(title = "Japan Yield Curve Spread (10Y-2Y)",
         xaxis = list(title = "Date"),
         yaxis = list(title = "Spread (%)"))
jp_plot
Code 
spread<- read_csv("../data/yield/merge_yield.csv")%>%
mutate(Date = as.Date(Date),
         Year = lubridate::year(Date)) %>%    
  arrange(Date)%>%
  mutate(across(ends_with("_spread"), ~ na.locf(.x, na.rm = FALSE))) %>%
  mutate(across(ends_with("_spread"), ~ na.locf(.x, fromLast = TRUE)))

eu_plot <- plot_ly(spread, x = ~Date, y = ~EU_Spread, type = 'scatter', mode = 'lines',
                   name = "EU (10Y-2Y)") %>%
  layout(title = "EU Yield Curve Spread (10Y-2Y)",
         xaxis = list(title = "Date"),
         yaxis = list(title = "Spread (%)"))
eu_plot
Code 
spread<- read_csv("../data/yield/merge_yield.csv")%>%
mutate(Date = as.Date(Date),
         Year = lubridate::year(Date)) %>%    
  arrange(Date)%>%
  mutate(across(ends_with("_spread"), ~ na.locf(.x, na.rm = FALSE))) %>%
  mutate(across(ends_with("_spread"), ~ na.locf(.x, fromLast = TRUE)))

uk_plot <- plot_ly(spread, x = ~Date, y = ~UK_Spread, type = 'scatter', mode = 'lines',
                   name = "UK (10Y-2Y)") %>%
  layout(title = "UK Yield Curve Spread (10Y-2Y)",
         xaxis = list(title = "Date"),
         yaxis = list(title = "Spread (%)"))
uk_plot
Code 
spread<- read_csv("../data/yield/merge_yield.csv")%>%
mutate(Date = as.Date(Date),
         Year = lubridate::year(Date)) %>%    
  arrange(Date)%>%
  mutate(across(ends_with("_spread"), ~ na.locf(.x, na.rm = FALSE))) %>%
  mutate(across(ends_with("_spread"), ~ na.locf(.x, fromLast = TRUE)))

eu_plot <- plot_ly(spread, x = ~Date, y = ~US_Spread, type = 'scatter', mode = 'lines',
                   name = "US (10Y-2Y)") %>%
  layout(title = "US Yield Curve Spread (10Y-2Y)",
         xaxis = list(title = "Date"),
         yaxis = list(title = "Spread (%)"))
eu_plot
  • FX rates:

all the currencies against USD has same patterns, clear upwards until 2008 and plumet after this period, with recovery period until 2020 and also decreases severly. in real economic situation, this reflects 2008 economic crisis and 2020 pandemic. eventually time series FX rate against USD shows impact of economic crisis. this also reflects FX rate has cyclic patterns. also for seasonality, it is hard to detect it.

  • Crude oil:

crude oil shows less movement relavantly to other assets. however, oil price also reflects economic situation, such as 2020 pandemic. one thing to note in crude oil price is that in terms of monthly, its movement has some kind of patterns.

  • S&P 500 index:

S&P500 graph showing upward trend as time goes by, specifically sharp upward trend for 2023. this plot as reflects economic situation, for example in 2025 first quarter, it decreases sharply, which this time, there was a tariff war, made investment slump due to economic & policy unstability.

  • Bitcoin:

Bitcoin is very similar as S&P500 but much radical movement, after 2023, it increases very sharply. it is hard to find seasonality or cycllic pattern for this data

  • yield curve spreads:

In yield curve spread( 10 year - 2 years ), three countries have similar pattern, but Japan has distinct pattern compare to other countries. Japan has increasing spread rate after 2020 whereas other countries has shifts. spreads generally a sign for economic policy expectation, as spread increases it could mean downward on inflation or better economic situation, whereas decrease could mean tightening from central bank, impacting investment and sign for bad economics.

Lag plots

Code 
jp_ts <- ts(jp$USD_JP, frequency = 252, start = c(year(min(jp$USD_JP))))
gglagplot(jp_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("JYP/USD")

Code 
uk_ts <- ts(uk$UK_USD, frequency = 252, start = c(year(min(uk$UK_USD))))
gglagplot(jp_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("UK/USD")

Code 
eu_ts <- ts(eu$EUR_USD, frequency = 252, start = c(year(min(eu$EUR_USD))))
gglagplot(jp_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("EU/USD")

Code 
crude_oil_ts <- ts(crude_oil$WTI, frequency = 365, start = c(year(min(crude_oil$WTI))))
gglagplot(jp_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("Crude oil price")

Code 
sp500_ts <- ts(sp500$Close, frequency = 252, start = c(year(min(sp500$Close))))
gglagplot(jp_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("S&P500 index ")

Code 
btc_ts <- ts(btc$Close, frequency = 252, start = c(year(min(btc$Close))))
gglagplot(jp_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("Bitcoin price")

Code 
jp_spread_ts <- ts(spread$JP_Spread, frequency = 252, start = c(year(min(spread$JP_Spread))))

gglagplot(jp_spread_ts, do.lines = FALSE) +
  labs(title = "Lag Plots of japan_spread",
       x = "Lags", y = "Japan spread %")

Code 
eu_spread_ts <- ts(spread$EU_Spread, frequency = 252, start = c(year(min(spread$EU_Spread))))

gglagplot(eu_spread_ts, do.lines = FALSE) +
  labs(title = "Lag Plots of euro_spread",
       x = "Lags", y = "Euro spread %")

Code 
uk_spread_ts <- ts(spread$UK_Spread, frequency = 252, start = c(year(min(spread$UK_Spread))))

gglagplot(uk_spread_ts, do.lines = FALSE) +
  labs(title = "Lag Plots of UK_spread",
       x = "Lags", y = "Uk spread %")

Code 
us_spread_ts <- ts(spread$US_Spread, frequency = 252, start = c(year(min(spread$US_Spread))))

gglagplot(us_spread_ts, do.lines = FALSE) +
  labs(title = "Lag Plots of US_spread",
       x = "Lags", y = "US spread %")

by observing Lag plots, all the variables has similar conclusion; it is strongly linear, which is strongly autocorrelated and non-stationary. Also it is hard to observe seasonality.

among plots, spread rate plots are less linear and autocorrelated compare to others, it has less centric to the line.

Decomposition

Code 
decompose_jp_ts <- decompose(jp_ts)
autoplot(decompose_jp_ts)

Code 
decompose_uk_ts <- decompose(uk_ts)
autoplot(decompose_uk_ts)

Code 
decompose_eu_ts <- decompose(eu_ts)
autoplot(decompose_eu_ts)

Code 
decompose_oil_ts <- decompose(crude_oil_ts)
autoplot(decompose_oil_ts)

Code 
decompose_sp500_ts <- decompose(sp500_ts)
autoplot(decompose_sp500_ts)

Code 
decompose_btc_ts <- decompose(btc_ts)
autoplot(decompose_btc_ts)

Code 
decompose_jp_spread_ts <- decompose(jp_spread_ts)
autoplot(decompose_jp_spread_ts)

Code 
decompose_eu_spread_ts <- decompose(eu_spread_ts)
autoplot(decompose_eu_spread_ts)

Code 
decompose_uk_spread_ts <- decompose(uk_spread_ts)
autoplot(decompose_uk_spread_ts)

Code 
decompose_us_spread_ts <- decompose(us_spread_ts)
autoplot(decompose_us_spread_ts)

For all the data, it has weak seasonlity as giving same conclusion as Lag plots.

ACF and PACF plots

Code 
ggAcf(jp_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of JPY currency rate") +
  theme_bw()

Code 
ggPacf(jp_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of JPY currency rate") +
  theme_bw()

Code 
#| echo: true
#| code-fold: true

ggAcf(uk_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of GBP currency rate") +
  theme_bw()

Code 
ggPacf(uk_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of GBP currency rate") +
  theme_bw()

Code 
#| echo: true
#| code-fold: true

ggAcf(eu_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of EUR currency rate") +
  theme_bw()

Code 
ggPacf(eu_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of EUR currency rate") +
  theme_bw()

Code 
#| echo: true
#| code-fold: true

ggAcf(crude_oil_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of Crude Oil Price") +
  theme_bw()

Code 
ggPacf(crude_oil_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of Crude Oil Price") +
  theme_bw()

Code 
#| echo: true
#| code-fold: true

ggAcf(sp500_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of S&P500 Index") +
  theme_bw()

Code 
ggPacf(sp500_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of S&P500 Index") +
  theme_bw()

Code 
ggAcf(btc_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of Bitcoin Price") +
  theme_bw()

Code 
ggPacf(btc_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of Bitcoin Price") +
  theme_bw()

Code 
ggAcf(jp_spread_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of Japan Spread Rate") +
  theme_bw()

Code 
ggPacf(jp_spread_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of Japan Spread Rate") +
  theme_bw()

Code 
ggAcf(eu_spread_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of Euro Spread Rate") +
  theme_bw()

Code 
ggPacf(eu_spread_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of Euro Spread Rate") +
  theme_bw()

Code 
ggAcf(us_spread_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of US Spread Rate") +
  theme_bw()

Code 
ggPacf(us_spread_ts) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of US Spread Rate") +
  theme_bw()

  • FX rates :

ACF plot shows slow decay and showing high ACF and PACF shows after lag 0-1, it signicantly falls to line, showing correlation like past results

  • crude oil, s&P500, bitcoin :

these data decays very radically, for crude oil and S&P500 it goes to negative value, which can say there is an inverse realtionship for observation k time step apart. for PACF, it also falls significantly after lag 0-1, showing correlation.

  • spread rate :

spread rate ACF also decays very radically, howevcer it doesn’t go to negative value except US spread rate, this might due that in time series plot, US rates are much radical compare to other country spread rate. for PACF plot, it can hold more lags; lag 2-3 or 4 is capable for spread rate plots, after that it is correlated. this result corresponds to lag plot since spread rate’s lag plot has less linear pattern compare to other data

Augmented Dickey-Fuller Test


    Augmented Dickey-Fuller Test

data:  jp_ts
Dickey-Fuller = -1.3225, Lag order = 18, p-value = 0.8653
alternative hypothesis: stationary
Code 
adf.test(uk_ts)

    Augmented Dickey-Fuller Test

data:  uk_ts
Dickey-Fuller = -2.2011, Lag order = 18, p-value = 0.4932
alternative hypothesis: stationary
Code 
adf.test(eu_ts)

    Augmented Dickey-Fuller Test

data:  eu_ts
Dickey-Fuller = -2.1396, Lag order = 18, p-value = 0.5192
alternative hypothesis: stationary
Code 
adf.test(crude_oil_ts)

    Augmented Dickey-Fuller Test

data:  crude_oil_ts
Dickey-Fuller = -2.2326, Lag order = 13, p-value = 0.4799
alternative hypothesis: stationary
Code 
adf.test(sp500_ts)

    Augmented Dickey-Fuller Test

data:  sp500_ts
Dickey-Fuller = -1.4167, Lag order = 10, p-value = 0.8252
alternative hypothesis: stationary
Code 
adf.test(btc_ts)

    Augmented Dickey-Fuller Test

data:  btc_ts
Dickey-Fuller = -1.3955, Lag order = 15, p-value = 0.8343
alternative hypothesis: stationary
Code 
adf.test(jp_spread_ts)

    Augmented Dickey-Fuller Test

data:  jp_spread_ts
Dickey-Fuller = -2.2361, Lag order = 14, p-value = 0.4784
alternative hypothesis: stationary
Code 
adf.test(eu_spread_ts)

    Augmented Dickey-Fuller Test

data:  eu_spread_ts
Dickey-Fuller = -1.2324, Lag order = 14, p-value = 0.9013
alternative hypothesis: stationary
Code 
adf.test(uk_spread_ts)

    Augmented Dickey-Fuller Test

data:  uk_spread_ts
Dickey-Fuller = -1.1484, Lag order = 14, p-value = 0.9148
alternative hypothesis: stationary
Code 
adf.test(us_spread_ts)

    Augmented Dickey-Fuller Test

data:  us_spread_ts
Dickey-Fuller = -1.1226, Lag order = 14, p-value = 0.919
alternative hypothesis: stationary

For all of categories, ADF test suggest it is non-statationary, since it has large p-value(0.05) which fails to reject null hypothesis. So, performing log transformation or differencing is well needed.

Differencing

Code 
ggAcf(diff(log(jp_ts))) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of JPY currency rate") +
  theme_bw()

Code 
ggPacf(diff(log(jp_ts))) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of JPY currency rate") +
  theme_bw()

Code 
#| echo: true
#| code-fold: true

ggAcf(diff(log(uk_ts))) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of GBP currency rate") +
  theme_bw()

Code 
ggPacf(diff(log(uk_ts))) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of GBP currency rate") +
  theme_bw()

Code 
#| echo: true
#| code-fold: true

ggAcf(diff(log(eu_ts))) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of GBP currency rate") +
  theme_bw()

Code 
ggPacf(diff(log(eu_ts))) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of GBP currency rate") +
  theme_bw()

Code 
#| echo: true
#| code-fold: true

ggAcf(log(crude_oil_ts)) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of Crude Oil Price") +
  theme_bw()

Code 
ggPacf(log(crude_oil_ts)) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of Crude Oil Price") +
  theme_bw()

Code 
#| echo: true
#| code-fold: true

ggAcf(diff(log(sp500_ts))) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of S&P500 Index") +
  theme_bw()

Code 
ggPacf(diff(log(sp500_ts))) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of S&P500 Index") +
  theme_bw()

Code 
ggAcf(diff(log(btc_ts))) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of Bitcoin Price") +
  theme_bw()

Code 
ggPacf(diff(log(btc_ts))) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of Bitcoin Price") +
  theme_bw()

Code 
ggAcf(diff(jp_spread_ts)) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of Japan Spread Rate") +
  theme_bw()

Code 
ggPacf(diff(jp_spread_ts)) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of Japan Spread Rate") +
  theme_bw()

Code 
ggAcf(diff(eu_spread_ts)) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of Euro Spread Rate") +
  theme_bw()

Code 
ggPacf(diff(eu_spread_ts)) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of Euro Spread Rate") +
  theme_bw()

Code 
ggAcf(diff(us_spread_ts)) +
  geom_segment(aes(xend = lag, yend = 0), color="blue") +
  labs(title = "ACF of US Spread Rate") +
  theme_bw()

Code 
ggPacf(diff(uk_spread_ts)) +
  geom_segment(aes(xend = lag, yend = 0), color="red") +
  labs(title = "PACF of US Spread Rate") +
  theme_bw()

for all the data, first differencing ACF and PACF plot shows similar patterns, which seems stationary. this will later on goes to ADF test.

ADF test after differencing


    Augmented Dickey-Fuller Test

data:  diff(log(jp_ts))
Dickey-Fuller = -18.43, Lag order = 18, p-value = 0.01
alternative hypothesis: stationary
Code 
adf.test(diff(log(uk_ts)))

    Augmented Dickey-Fuller Test

data:  diff(log(uk_ts))
Dickey-Fuller = -18.859, Lag order = 18, p-value = 0.01
alternative hypothesis: stationary
Code 
adf.test(diff(log(eu_ts)))

    Augmented Dickey-Fuller Test

data:  diff(log(eu_ts))
Dickey-Fuller = -18.245, Lag order = 18, p-value = 0.01
alternative hypothesis: stationary
Code 
adf.test(diff(crude_oil_ts))

    Augmented Dickey-Fuller Test

data:  diff(crude_oil_ts)
Dickey-Fuller = -15.053, Lag order = 13, p-value = 0.01
alternative hypothesis: stationary
Code 
adf.test(diff(log(sp500_ts)))

    Augmented Dickey-Fuller Test

data:  diff(log(sp500_ts))
Dickey-Fuller = -11.132, Lag order = 10, p-value = 0.01
alternative hypothesis: stationary
Code 
adf.test(diff(log(btc_ts)))

    Augmented Dickey-Fuller Test

data:  diff(log(btc_ts))
Dickey-Fuller = -14.791, Lag order = 15, p-value = 0.01
alternative hypothesis: stationary
Code 
adf.test(diff(jp_spread_ts))

    Augmented Dickey-Fuller Test

data:  diff(jp_spread_ts)
Dickey-Fuller = -14.815, Lag order = 14, p-value = 0.01
alternative hypothesis: stationary
Code 
adf.test(diff(eu_spread_ts))

    Augmented Dickey-Fuller Test

data:  diff(eu_spread_ts)
Dickey-Fuller = -15.106, Lag order = 14, p-value = 0.01
alternative hypothesis: stationary
Code 
adf.test(diff(uk_spread_ts))

    Augmented Dickey-Fuller Test

data:  diff(uk_spread_ts)
Dickey-Fuller = -13.881, Lag order = 14, p-value = 0.01
alternative hypothesis: stationary

    Augmented Dickey-Fuller Test

data:  diff(us_spread_ts)
Dickey-Fuller = -13.978, Lag order = 14, p-value = 0.01
alternative hypothesis: stationary

for all the data, compare to ADF test before differencing, now rejects null hypothesis due to small p- value, concluding it is stationary

Moving average

Code 
jp_df <- data.frame(Date = time(jp_ts), Value = as.numeric(jp_ts))
jp_df$MA20 <- rollmean(jp_df$Value, 20, fill = NA, align = "right")
jp_df$MA50 <- rollmean(jp_df$Value, 50, fill = NA, align = "right")
jp_df$MA100 <- rollmean(jp_df$Value, 100, fill = NA, align = "right")

plot_ly(jp_df, x = ~Date) %>%
  add_lines(y = ~Value, name = "JPY Rate", line = list(color = "black")) %>%
  add_lines(y = ~MA20, name = "MA20", line = list(color = "blue")) %>%
  add_lines(y = ~MA50, name = "MA50", line = list(color = "red")) %>%
  add_lines(y = ~MA100, name = "MA100", line = list(color = "green")) %>%
  layout(title = "JPY Currency Rate with Moving Averages",
         xaxis = list(title = "Date"),
         yaxis = list(title = "JPY rate"))
Code 
uk_df <- data.frame(Date = time(uk_ts), Value = as.numeric(uk_ts))
uk_df$MA20 <- rollmean(uk_df$Value, 20, fill = NA, align = "right")
uk_df$MA50 <- rollmean(uk_df$Value, 50, fill = NA, align = "right")
uk_df$MA100 <- rollmean(uk_df$Value, 100, fill = NA, align = "right")

plot_ly(uk_df, x = ~Date) %>%
  add_lines(y = ~Value, name = "GBP Rate", line = list(color = "black")) %>%
  add_lines(y = ~MA20, name = "MA20", line = list(color = "blue")) %>%
  add_lines(y = ~MA50, name = "MA50", line = list(color = "red")) %>%
  add_lines(y = ~MA100, name = "MA100", line = list(color = "green")) %>%
  layout(title = "GBP Currency Rate with Moving Averages",
         xaxis = list(title = "Date"),
         yaxis = list(title = "GBP rate"))
Code 
eu_df <- data.frame(Date = time(eu_ts), Value = as.numeric(eu_ts))
eu_df$MA20 <- rollmean(eu_df$Value, 20, fill = NA, align = "right")
eu_df$MA50 <- rollmean(eu_df$Value, 50, fill = NA, align = "right")
eu_df$MA100 <- rollmean(eu_df$Value, 100, fill = NA, align = "right")

plot_ly(eu_df, x = ~Date) %>%
  add_lines(y = ~Value, name = "EUR Rate", line = list(color = "black")) %>%
  add_lines(y = ~MA20, name = "MA20", line = list(color = "blue")) %>%
  add_lines(y = ~MA50, name = "MA50", line = list(color = "red")) %>%
  add_lines(y = ~MA100, name = "MA100", line = list(color = "green")) %>%
  layout(title = "EUR Currency Rate with Moving Averages",
         xaxis = list(title = "Date"),
         yaxis = list(title = "EUR rate"))
Code 
oil_df <- data.frame(Date = time(crude_oil_ts), Value = as.numeric(crude_oil_ts))
oil_df$MA20 <- rollmean(oil_df$Value, 20, fill = NA, align = "right")
oil_df$MA50 <- rollmean(oil_df$Value, 50, fill = NA, align = "right")
oil_df$MA100 <- rollmean(oil_df$Value, 100, fill = NA, align = "right")

plot_ly(oil_df, x = ~Date) %>%
  add_lines(y = ~Value, name = "Oil Price", line = list(color = "black")) %>%
  add_lines(y = ~MA20, name = "MA20", line = list(color = "blue")) %>%
  add_lines(y = ~MA50, name = "MA50", line = list(color = "red")) %>%
  add_lines(y = ~MA100, name = "MA100", line = list(color = "green")) %>%
  layout(title = "Crude Oil Price with Moving Averages",
         xaxis = list(title = "Date"),
         yaxis = list(title = "Oil Price"))
Code 
sp500_df <- data.frame(Date = time(sp500_ts), Value = as.numeric(sp500_ts))
sp500_df$MA20 <- rollmean(sp500_df$Value, 20, fill = NA, align = "right")
sp500_df$MA50 <- rollmean(sp500_df$Value, 50, fill = NA, align = "right")
sp500_df$MA100 <- rollmean(sp500_df$Value, 100, fill = NA, align = "right")

plot_ly(sp500_df, x = ~Date) %>%
  add_lines(y = ~Value, name = "S&P500", line = list(color = "black")) %>%
  add_lines(y = ~MA20, name = "MA20", line = list(color = "blue")) %>%
  add_lines(y = ~MA50, name = "MA50", line = list(color = "red")) %>%
  add_lines(y = ~MA100, name = "MA100", line = list(color = "green")) %>%
  layout(title = "S&P500 Index with Moving Averages",
         xaxis = list(title = "Date"),
         yaxis = list(title = "S&P500"))
Code 
btc_df <- data.frame(Date = time(btc_ts), Value = as.numeric(btc_ts))
btc_df$MA20 <- rollmean(btc_df$Value, 20, fill = NA, align = "right")
btc_df$MA50 <- rollmean(btc_df$Value, 50, fill = NA, align = "right")
btc_df$MA100 <- rollmean(btc_df$Value, 100, fill = NA, align = "right")

plot_ly(btc_df, x = ~Date) %>%
  add_lines(y = ~Value, name = "BTC Price", line = list(color = "black")) %>%
  add_lines(y = ~MA20, name = "MA20", line = list(color = "blue")) %>%
  add_lines(y = ~MA50, name = "MA50", line = list(color = "red")) %>%
  add_lines(y = ~MA100, name = "MA100", line = list(color = "green")) %>%
  layout(title = "Bitcoin Price with Moving Averages",
         xaxis = list(title = "Date"),
         yaxis = list(title = "BTC Price"))
Code 
jp_spread_df <- data.frame(Date = time(jp_spread_ts), Value = as.numeric(jp_spread_ts))
jp_spread_df$MA20 <- rollmean(jp_spread_df$Value, 20, fill = NA, align = "right")
jp_spread_df$MA50 <- rollmean(jp_spread_df$Value, 50, fill = NA, align = "right")
jp_spread_df$MA100 <- rollmean(jp_spread_df$Value, 100, fill = NA, align = "right")

plot_ly(jp_spread_df, x = ~Date) %>%
  add_lines(y = ~Value, name = "JP Spread", line = list(color = "black")) %>%
  add_lines(y = ~MA20, name = "MA20", line = list(color = "blue")) %>%
  add_lines(y = ~MA50, name = "MA50", line = list(color = "red")) %>%
  add_lines(y = ~MA100, name = "MA100", line = list(color = "green")) %>%
  layout(title = "Japan Spread with Moving Averages",
         xaxis = list(title = "Date"),
         yaxis = list(title = "Spread (bps)"))
Code 
eu_spread_df <- data.frame(Date = time(eu_spread_ts), Value = as.numeric(eu_spread_ts))
eu_spread_df$MA20 <- rollmean(eu_spread_df$Value, 20, fill = NA, align = "right")
eu_spread_df$MA50 <- rollmean(eu_spread_df$Value, 50, fill = NA, align = "right")
eu_spread_df$MA100 <- rollmean(eu_spread_df$Value, 100, fill = NA, align = "right")

plot_ly(eu_spread_df, x = ~Date) %>%
  add_lines(y = ~Value, name = "EU Spread", line = list(color = "black")) %>%
  add_lines(y = ~MA20, name = "MA20", line = list(color = "blue")) %>%
  add_lines(y = ~MA50, name = "MA50", line = list(color = "red")) %>%
  add_lines(y = ~MA100, name = "MA100", line = list(color = "green")) %>%
  layout(title = "Euro Spread with Moving Averages",
         xaxis = list(title = "Date"),
         yaxis = list(title = "Spread (bps)"))
Code 
uk_spread_df <- data.frame(Date = time(uk_spread_ts), Value = as.numeric(uk_spread_ts))
uk_spread_df$MA20 <- rollmean(uk_spread_df$Value, 20, fill = NA, align = "right")
uk_spread_df$MA50 <- rollmean(uk_spread_df$Value, 50, fill = NA, align = "right")
uk_spread_df$MA100 <- rollmean(uk_spread_df$Value, 100, fill = NA, align = "right")

plot_ly(uk_spread_df, x = ~Date) %>%
  add_lines(y = ~Value, name = "UK Spread", line = list(color = "black")) %>%
  add_lines(y = ~MA20, name = "MA20", line = list(color = "blue")) %>%
  add_lines(y = ~MA50, name = "MA50", line = list(color = "red")) %>%
  add_lines(y = ~MA100, name = "MA100", line = list(color = "green")) %>%
  layout(title = "UK Spread with Moving Averages",
         xaxis = list(title = "Date"),
         yaxis = list(title = "Spread (bps)"))
Code 
us_spread_df <- data.frame(Date = time(us_spread_ts), Value = as.numeric(us_spread_ts))
us_spread_df$MA20 <- rollmean(us_spread_df$Value, 20, fill = NA, align = "right")
us_spread_df$MA50 <- rollmean(us_spread_df$Value, 50, fill = NA, align = "right")
us_spread_df$MA100 <- rollmean(us_spread_df$Value, 100, fill = NA, align = "right")

plot_ly(us_spread_df, x = ~Date) %>%
  add_lines(y = ~Value, name = "US Spread", line = list(color = "black")) %>%
  add_lines(y = ~MA20, name = "MA20", line = list(color = "blue")) %>%
  add_lines(y = ~MA50, name = "MA50", line = list(color = "red")) %>%
  add_lines(y = ~MA100, name = "MA100", line = list(color = "green")) %>%
  layout(title = "US Spread with Moving Averages",
         xaxis = list(title = "Date"),
         yaxis = list(title = "Spread (bps)"))

three MA: 20,50 ,100

for all the plots, 50 moving average aligns well with data, for 100 moving average, it seems like it is overfitting.